Answer
exist for a ∈(−∞,0)∪(0,∞)
Work Step by Step
If we reach the same point by approaching from either the left or right direction, then $\lim\limits_{x \to a} f(x)$ exists.
From the graph above, we can see that the above description is true for all points but $a=0$.
When we approach $x=0$ from the left/negative direction, the $y$ coordinate approaches $1$. therefore $\lim\limits_{x \to 0^{-}} f(x)=1$.
When we approach $x=0$ from the right/positive direction, the $y$ coordinate approaches $-1$. therefore $\lim\limits_{x \to 0^{+}} f(x)=-1$.
Since $\lim\limits_{x \to 0^{-}} f(x)\ne \lim\limits_{x \to 0^{+}} f(x)$, the following limit does not exist: $\lim\limits_{x \to 0} f(x)$.
